Geodesic Rays and Kähler–Ricci Trajectories on Fano
Manifolds
Tamás Darvas∗ and Weiyong He†
arXiv:1411.0774v1 [math.DG] 4 Nov 2014
Abstract
Suppose (X, J, ω) is a Fano manifold and t → rt is a diverging Kähler-Ricci
trajectory. We construct a bounded geodesic ray t → ut weakly asymptotic to
t → rt , along which Ding’s F–functional decreases. In absence of non-trivial
holomorphic vector fields this proves the equivalence between geodesic stability
of the F–functional and existence of Kähler–Einstein metrics. We also explore
applications of our construction to Tian’s α–invariant.
1
Introduction and Main Results
We consider a Fano manifold (X, J, ω). The space of smooth Kähler potentials H is the
set
¯ > 0}.
H = {u ∈ C ∞ (X)| ωu := ω + i∂ ∂u
Clearly, H is a Fréchet manifold as an open subset of C ∞ (X). For v ∈ H one can identify
Tv H with C ∞ (X). Given 1 ≤ p < ∞, one can introduce a Finsler-metric on H:
Z
1
p1
(1)
kξkp,u =
|u|pωun , ξ ∈ Tv H,
Vol(X) X
R
where Vol(X) = X ω n . This is a generalization of the Mabuchi Riemannian metric initially investigated in [Ma, Se, Do], which corresponds to the case p = 2. This Finsler
structure (along with a more general situation involving Orlicz norms) was studied extensively in [Da4]. Below and in Section 2.1 we summarize the results that we will need
the most from this work. As usual, the length of a smooth curve [0, 1] ∋ t → αt ∈ H is
computed by the formula:
Z
1
lp (α) =
kα̇t kp,αt dt.
(2)
0
The path length distance dp (u0 , u1) between u0 , u1 ∈ H is the infimum of the length
of smooth curves joining u0, u1 . In [Da4] it is proved that dp (u0 , u1) = 0 if and only if
u0 = u1 , thus (H, dp ) is a metric space, which is a generalization of a result of X.X. Chen
in the case p = 2 [C].
Let us recall some facts about geodesic segments in the Riemannian case p = 2.
Suppose S = {0 < Re s < 1} ⊂ C. Following [Se], one can compute that a smooth
∗
Research supported by BSF grant 2012236.
Research supported partially by NSF grant 1005392.
2010 Mathematics subject classification 53C55, 32W20, 32U05.
†
1
curve [0, 1] ∋ t → ut ∈ H connecting u0 , u1 ∈ H is a geodesic if its complexification
u(s, x) = uRe s (x) is the (unique) smooth solution of the following Dirichlet problem on
S × X:
(π ∗ ω + i∂∂u)n+1 = 0
u(t + ir, x) = u(t, x) ∀x ∈ X, t ∈ (0, 1), r ∈ R
u(0, x) = u0 (x), u(1, x) = u1 (x), x ∈ X.
(3)
Unfortunately the above problem does not have smooth solutions (see [LV, Da3]), but
¯ has bounded
a unique solution in the sense of Bedford-Taylor does exist such that i∂ ∂u
coefficients (see [C] with complements in [Bl]). The most general result about regularity
was proved in [BD, Brm2] (see [H1] for a different approach) but regularity higher then
C 1,α is not possible by examples provided in [DL].
The curve t → ut is called the weak geodesic joining u0 , u1 . As argued in [Da4], this
same curve interacts well with all the metrics dp , i.e.
dp (u0 , u1) = ku̇t kp,ut , t ∈ [0, 1], p ≥ 1,
(4)
and t → ut is an actual metric geodesic joining u0 , u1 in the metric completion (H, dp ) =
(E p (X, ω), dp ) that we introduce now.
Given u ∈ PSH(X, ω), as explained in [GZ1], one can define the non-pluripolar
measure ωun that coincides with the usual Bedford-Taylor
when u is bounded.
R n R volume
n
n
We say that ωu has full volume (u ∈ E(X, ω)) if X ωu = X ω . Given v ∈ E(X, ω), we
say that v ∈ E p (X, ω) if
Z
Ep (v) =
X
|v|p ωvn < ∞,
For a quick review of finite energy classes we refer to [Da3, Section 2.3].
Next we introduce a geodesic metric space structure on E p (X, ω), following [Da4].
Suppose u0 , u1 ∈ E p (X, ω). Let {uk0 }k∈N , {uk1 }k∈N ⊂ H be sequences decreasing pointwise
to u0 and u1 respectively. By [BK, De] it is always possible to find such approximating
sequences. We define the metric dp (u0, u1 ) as follows:
dp (u0 , u1 ) = lim dp (uk0 , uk1 ).
k→∞
(5)
As justified in [Da4, Theorem 2] the above limit exists, is well defined, and dp (u0 , u1 ) = 0
implies u0 = u1 . Let us also define geodesics in this space. Let
ukt : [0, 1] → H∆ := PSH(X, ω) ∩ {∆u ∈ L∞ }
be the weak geodesic joining uk0 , uk1 . We define t → ut as the decreasing limit:
ut = lim ukt , t ∈ (0, 1).
k→+∞
(6)
The curve t → ut is well defined and ut ∈ E p (X, ω), t ∈ (0, 1), as follows from [Da3,
Theorem 6]. By [Da4, Theorem 2] this curve is a dp –geodesic joining u0 , u1 and we have
(H, dp ) = (E p (X, ω), dp), p ≥ 1.
2
(7)
Recall that by a ρ–geodesic in a metric space (M, ρ) we understand a curve (a, b) ∋
t → gt ∈ M for which there exists C > 0 satisfying:
ρ(gt1 , gt2 ) = C|t1 − t2 |, t1 , t2 ∈ (a, b).
′
By the definition, we have the inclusion E p (X, ω) ⊂ E p (X, ω), for p′ ≤ p and also the
metric dp dominates dp′ . What is more, it follows that for u0 , u1 ∈ E p (X, ω), the curve
defined in (6) is a geodesic with respect to both dp and d′p (perhaps of different length).
Lastly, by the definition of the finite energy classes we have the inclusion
\
H0 = PSH(X, ω) ∩ L∞ (X) ⊂
E p (X, ω).
(8)
p≥1
By the above, for u0 , u1 ∈ H0 , the curve (0, 1) ∋ t → ut ∈ H0 from (6) will be a dp –
geodesic joining u0 , u1 for all p ≥ 1. This observation will be crucial in future arguments.
Functionals play an important role in the investigation of special Kähler metrics.
Recall that the Aubin-Mabuchi energy and Ding’s F –functional are defined as follows:
n
X
1
AM(v) =
(n + 1)Vol(X) j=0
Z
¯ n−j ,
vω j ∧ (ω + i∂ ∂v)
(9)
X
F (v) = −AM(v) − log
Z
e−v+fω ω n ,
(10)
X
¯ ω
where v ∈ H and fω ∈ C ∞ (X) is the Ricci potential of ω, i.e. Ric ω = ω + i∂ ∂f
normalized by
Z
efω ω n = 1.
X
It was proved in [Da4] that both of these functionals are continuous with respect to all
metrics dp , hence extend to E p (X, ω) continuously. As the map u → ωu is translation
invariant one may want normalize Kähler potentials to obtain an equivalence between
metrics and potentials. This can be done by only considering potentials from the ”totally
geodesic” hypersurfaces
HAM = H ∩ {AM(·) = 0},
H0,AM = L∞ (X) ∩ PSH(X, ω) ∩ {AM(·) = 0},
p
EAM
(X, ω) = E p (X, ω) ∩ {AM(·) = 0}.
A smooth metric ωuKE is Kähler-Einstein if ωuKE = Ric ωuKE . One can study such
metrics by looking at the long time asymptotics of the Hamilton’s Kähler–Ricci flow:
( dω
rt
= −Ric ωrt + ωrt ,
dt
(11)
r0 = v.
As proved by Cao [Cao], for any v ∈ HAM , this PDE has a smooth solution [0, 1) ∋ t →
rt ∈ HAM . It follows from a theorem of Perelman and work of Chen-Tian, Tian-Zhu and
Phong-Song-Sturm-Weinkove, that whenever a Kähler–Einstein metric cohomologous
to ω exists, then ωrt converges exponentially fast to one such metric (see [CT], [TZ],
[PSSW]).
3
We remark that our choice of normalization is different from the alternatives used
in the literature (see [BEG, Chapter 6]). We choose to work with the normalization
AM(·) = 0, as this seems to be the most natural one from the point of view of Mabuchi
geometry. Indeed, that Aubin-Mabuchi energy is continuous with respect to all metrics
dp and is linear along the geodesic segments defined in (6). It will require some careful
analysis, but as we shall see, from the point of view of long time asymptotics, this
normalization is equivalent to other alternatives.
Suppose (M, ρ) is a geodesic metric space and [0, ∞) ∋ t → ct ∈ M is a continuous
curve. We say that the unit speed ρ–geodesic ray [0, ∞) ∋ t → gt ∈ M is weakly
asymptotic to the curve t → ct , if there exists tj → ∞ for which there exist ρ–geodesic
segments [0, ρ(c0 , ctj )] ∋ t → gtj ∈ M connecting c0 and ctj such that
lim ρ(gtj , gt ) = 0, t ∈ [0, ∞).
j→∞
We clearly need limj ρ(c0 , ctj ) = ∞ in this last definition, hence to construct dp –
geodesic rays weakly asymptotic to diverging Kähler-Ricci trajectories, we first need to
prove the following result, which generalizes [Da4, Theorem 6] and the main result of
[Mc].
Theorem 1. Suppose (X, J, ω) is a Fano manifold and p ≥ 1. There exists a Kähler–
Einstein metric in H if and only if every Kähler–Ricci trajectory [0, ∞) ∋ t → rt ∈ HAM
is dp –bounded.
Using this theorem, the main result of [BrmBrn], the compactness theorem of [BBEGZ]
and the divergence analysis of Kähler-Ricci trajectories from [R1], we can establish our
main result:
Theorem 2. Suppose (X, J, ω) is a Fano manifold without a Kähler–Einstein metric in
H and [0, ∞) ∋ t → rt ∈ HAM is a Kähler-Ricci trajectory. Then there exists a curve
[0, ∞) ∋ t → ut ∈ H0,AM which is a dp –geodesic ray weakly asymptotic to t → rt for all
p ≥ 1. In addition to this, t → ut satisfies the following:
(i) t → F (ut ) is decreasing,
(ii) the ”sup-normalized” potentials
unt − supX (ut − u0 ) ∈ H0 decrease pointwise to
R
u∞ ∈ PSH(X, ω) for which X e− n+1 u∞ ω n = ∞.
If additionally (X, J) does not admit non–trivial holomorphic vector fields then t → F (ut )
is strictly decreasing.
We note that the normalizing condition AM(ut ) = 0 in the above result assures that
geodesic ray t → ut is non–trivial, i.e. ut 6= u0 + ct.
This theorem provides a partial answer to a folklore conjecture, perhaps first suggested by [LNT], which says that one should be able to construct ”destabilizing” geodesic
rays (strongly) asymptotic to diverging Kähler-Ricci trajectories. For a precise statement
and connections with other results we refer to [R1, Conjecture 4.10].
We hope that the methods developed here will be the building blocks of future results
constructing geodesic rays asymptotic to different (geometric) flow trajectories. We refer
to Theorem 3.4 for a general result in this direction.
4
On Fano manifolds not admitting Kähler-Einstein metrics, condition (ii) above ensures the bound α(X) ≤ n/(n + 1) for Tian’s alpha invariant:
n Z
o
α(X) = sup α,
e−α(u−supX u) ω n ≤ Cα < +∞, u ∈ P SH(X, ω) .
X
This is a well known result of Tian
The fact that the geodesic ray t → ut is able to
R [T1].
n
detect a potential u∞ satisfying X e− n+1 u ω n = ∞, is analogous to results of [R1], where
it is shown that one can find such potential using a sequence of metrics along a diverging
Kähler-Ricci trajectory as well. We refer to this paper for relations with Nadel sheaves.
It would be interesting to see if the geodesic ray produced by the above theorem is in
fact unique. We prove that this ray is bounded, but it is not clear if this curve has more
regularity. Finally, we believe that t → F (ut ) is strictly decreasing regardless whether
(X, J) admits non–trivial holomorphic vector fields or not.
Finally, we note the following theorem, which is a consequence of the above result,
and in the case p = 2 gives the Kähler-Einstein analog of Donaldson’s conjectures on
existence of constant scalar curvature metrics [Do, H2]:
Theorem 3. Suppose p ∈ {1, 2} and (X, J, ω) is a Fano manifold without non–trivial
holomorphic vector–fields and u ∈ H. There exists no Kähler-Einstein metric in H if
and only if for any u0 ∈ H there exists a dp –geodesic ray [0, ∞) ∋ t → ut ∈ H0,AM with
u0 = u such that the function t → F (ut ) is strictly decreasing.
Proof. The only if direction is a consequence of the previous theorem. Now we argue
the if direction. Suppose there exists a Kähler–Einstein metric in H. In case p = 1 it
is enough to invoke [Da4, Theorem 6]. Indeed, this result says that on a Fano manifold
without non–trivial holomorphic vector–fields existence of a Kähler-Einstein metric in H
is equivalent to the d1 –properness of F (sublevel sets of F are d1 –bounded). Hence the
convex map t → F (ut) is eventually strictly increasing for any d1 –geodesic ray t → ut .
The case p = 2 follows if one notices that d2 –geodesic rays are also d1 –geodesic rays.
Indeed, this follows from the CAT (0) property of (H, d2 ) = (E 2(X, ω), d2 ) (see [Da3,
Theorem 6(iii)], [CC]). Because of this, d2 –geodesic segments connecting different points
of (E 2(X, ω), d2) are unique, hence they are always of the type described in (6), which are
also d1 –geodesics in (E 1 (X, ω), d1) (as remarked after (7)). The same statement holds
for geodesic rays as well, not just segments. Now we can use [Da4, Theorem 6(iii)] again
to conclude the argument.
We note here that for p = 2 this last theorem follows from the work of Berman
on K-polystability [Brm1]. Our approach however is purely analytical and avoids the
use of the recently established equivalence between K–stability and existence of Kähler–
Einstein metrics.
Although we do not pursue such generality, we remark that Theorem 1 and Theorem
2 also hold for the very general Orlicz-Finsler structures (H, dχ ) studied in [Da4].
Acknowledgements. The first author would like to thank Yanir Rubinstein for numerous
stimulating conversations related to the topic of the paper and for László Lempert for
suggestions on how to improve the presentation.
5
2
2.1
Preliminaries
The Metric Spaces (H, dp)
In hopes of characterizing convergence in (E p (X, ω), dp ) more explicitly, for u0 , u1 ∈
E p (X, ω) one introduces the following functional (see [Da4, G]):
1/p Z
Z
1/p
p n
p n
Ip (u0 , u1 ) =
|u0 − u1 | ωu0
+
|u0 − u1 | ωu1
.
X
X
In [Da4, Theorem 3] it is proved that there exists C(p) > 1 such that
1
Ip (u0 , u1 ) ≤ dp (u0 , u1 ) ≤ CIp (u0 , u1 ).
C
(12)
This double estimate implies that there exists C(p) > 1 such that
sup u ≤ Cdp (u, 0) + C.
X
Also, if dp (uk , u) → 0 then uk → u a.e. and also ωunk → ωun weakly. For more details we
refer to [Da4, Theorems 3-6]. We also note the following:
Proposition 2.1. Suppose {uk }k∈N ⊂ H0 = PSH(X, ω) ∩ L∞ and kuk kL∞ ≤ D for some
D > 0. Then {uk }k∈N is dp –Cauchy if and only if it is d1 –Cauchy. If this condition holds
then in addition the limit u = limk uk also satisfies kukL∞ ≤ D.
Proof. The equivalence follows from (12) and basic facts about Lp norms. The estimate
kukL∞ ≤ D also follows, as from [Da4, Theorem 5(i)] we have uk → u in capacity, hence
uk → u pointwise a.e..
We recall the compactness theorem [BBEGZ, Theorem 2.17]. Before we write down
the statement, let us first recall the notion of strong convergence and and entropy.
As introduced in [BBEGZ], we say that a sequence uk ∈ H converges strongly to
u ∈ E 1 (X, ω) if uk →L1 u and AM(uk ) → AM(u). The Mabuchi K-energy functional
M : H → R is given by the following formula:
M(u) = R̄AM(u) − L(u) + Hω (ωu ),
R
R
where R̄ = X Rω ∧ω n is the average scalar curvature Rω of ω, Hω (ωu ) = X log(ωun /ω n )ωun
is the entropy of ωun with respect to ω n and L(u) is the following operator:
L(u) =
n−1 Z
X
j=0
X
u Ric ω ∧ ωuj ∧ ω n−1−j .
Proposition 2.2. [BBEGZ, Proposition 2.6, Theorem 2.17] Suppose {uk }k∈N ⊂ H is
such that | supX uk |, Hω (ωuk ) ≤ D for some D ≥ 0. Then there exists u ∈ E 1 (X, ω) and
kl → ∞ such that ukl → u strongly.
It follows from the results of [BBGZ, BBEGZ] that one has uk → u strongly if and
only if I1 (uk , u) → 0, which in turn is equivalent to d1 (uk , u) → 0 according to (12).
Putting together the last two results we can write:
6
Theorem 2.3. Suppose {uk }k∈N ⊂ H is such that Hω (ωuk ), kuk kL∞ ≤ D for some
D ≥ 0. Then there exists u ∈ H0 with kukL∞ ≤ D and kl → ∞ such that dp (ukl , u) → 0
for all p ≥ 1.
In our computations we will need the following bound for the L functional in the
expression of the Mabuchi K-energy:
Proposition 2.4. For any p ≥ 1 there exists C(p) > 1 such that
|L(u)| ≤ Cdp (0, u), u ∈ H.
Proof. There exists C > 0 such that Ric ω ≤ Cω. We can start writing:
n Z
X
Z u n
|u|ω ∧
≤C
|L(u)| ≤ C
ω u2
X 2
j=1 X
Z u p 1/p
u
n
≤ Cdp (0, u),
≤C
≤ Cdp 0,
ω u2
2
X 2
j
ωun−j
where in the penultimate inequality we have used (12) and in the last inequality we have
used [Da4, Lemma 5.3].
Finally, we recall a result about bounded geodesics which will be very useful to us
later:
Theorem 2.5. [Da2, Theorem 1] Given a bounded weak geodesic [0, 1] ∋ t → ut ∈ H0
connecting u0 , u1 ∈ H0 , i.e. a bounded solution to (3), there exists Mu , mu ∈ R such that
for any a, b ∈ [0, 1] we have
(i) inf X
(ii) supX
ua −ub
a−b
ua −ub
a−b
= mu ,
= Mu .
This result tells us that for a bounded weak geodesic [0, 1] ∋ t → ut ∈ H0 the
function t → supX (ut − u0 ) is linear. As explained in [Da2], this implies that t → ũt =
ut − supX (ut − u0 ) ∈ H is a geodesic that is decreasing in t (one can see that ũ˙ t ≤ 0).
Clearly, supX ũt is bounded, hence the pointwise limit u∞ = limt→∞ ũt is different from
−∞. As we shall see by the end of this paper, for certain geodesic rays one can draw
geometric conclusions by studying the singularity type of u∞ .
2.2
Diverging Kähler-Ricci Trajectories
In this short section we recall some results from [R2], where a careful analysis of diverging Kähler-Ricci trajectories has been carried out. Unfortunately, this work uses a
normalization different from ours, but here we argue that the most important estimates
have analogs for ”AM–normalized” trajectories as well.
It is well known that flow equation (11) can be rewritten as the scalar equation as
n
ωrt = efω −rt +ṙt +β(t) ω n , where β : [0, ∞) → R is a function chosen depending on the
desired normalization condition on rt . In our investigations we will eventually use the
condition AM(rt ) = 0, however most of the literature on the Kähler-Ricci flow uses a
different normalization (that we will denote by t → r̃t ) for which β(t) = 0 and r̃0 = v + c,
7
with c carefully chosen (see [PSS, (2.10)]). Consequently, in this case the scalar equation
becomes
˙
ωr̃nt = e−r̃t +r̃t +fω ω n ,
(13)
and the conversion from this normalization to the one employed by us is given by the
formula
rt = r̃t − AM(r̃t ), t ≥ 0.
The following result brings together estimates for the trajectory t → r̃t that we will need
the most. Most of these are classical and well known, for the others we give a proof:
Proposition 2.6. Suppose t → r̃t is a Kähler-Ricci trajectory normalized as discussed
above. For any t ≥ 0 we have:
(i) kr̃˙t kL∞ , kfωt kL∞ ≤ C for some C > 1.
R
R
(ii) −C ≤ AM(r̃t ), in particular − X r̃t ωr̃nt ≤ n X r̃t ω n + C for some C > 1.
R
R
(iii) X r̃t ωr̃nt ≤ C, −C ≤ X r̃t ω n hence also −C ≤ supX r̃t for some C > 1.
(iv) − inf X r̃t ≤ C supX r̃t + D for some C, D > 0.
R
(v) − log X e−α(r̃t −supX r̃t ) ω n ≤ ((1 − α)n − α) supX r̃t + C for some C > 1.
(vi) supX r̃t −AM(r̃t ) ≥ supX r̃t /C −C ≥ (AM(r̃t )−inf X r̃t )/D−D for some C, D > 1.
Proof. The estimates in (i) are essentially due to Perelman [ST, TZ]. The estimates from
(ii) are also well known. We recall the argument from [R2]. First we notice that
Z
Z
˙
−r̃t +fω n
− log
e
ω = − log
e−r̃t ωr̃nt ,
X
X
hence this quantity is uniformly bounded by (i). It is well known that t → F (r̃t ) is
decreasing and now looking at the expression of F (r̃t ) from (10), we conclude that there
exists C > 1 such that AM(r̃t ) ≥ −C. The second estimate of (ii) now follows from the
next well known inequality:
n
X
1
AM(r̃t ) =
(n + 1)Vol(X) j=0
Z
X
j
r̃t ω ∧
ωr̃n−j
t
1
≤
(n + 1)Vol(X)
Z
X
r̃t ωr̃nt
+n
Z
X
r̃t ω .
n
R
R ˙
We now prove the estimate of
From (13) we have X er̃t ωr̃nt = X er̃t +f ω n . Hence
R (iii).
the estimates of (i) yield that X er̃t ωr̃nt is uniformly bounded. The first estimate now
follows from Jensen’s inequality:
Z
Z
1
1
n
r̃t ωr̃t ≤ log
er̃t ωr̃nt .
Vol(X) X
Vol(X) X
The second and third estimate of (iii) follows now from (ii). Estimate (iv) is just the
Harnack estimate for the Kähler-Ricci flow. For a summary of the proof we refer to steps
(i) and (iii) in the proof of [R2, Theorem 1.3], which in turn follows [T1].
8
We justify the estimate of (v) and the roots of our argument are again from [R2]. To
start, we notice that using equation (13) we can write
Z
Z
˙
−αr̃t n
e−αr̃t +r̃t −fω +r̃t ωr̃nt
− log
e
ω = − log
X
ZX
Z
1
n(1 − α)
n
≤
(α − 1)r̃t ωr̃t + C ≤
r̃t ω n + C,
Vol(X) X
Vol(X) X
where in the second line we have used the estimates of (i) and (ii). This finishes the
proof of (v).
Now we turn to the proof of the double estimate in (vi). From the definition of AM
and (iii) it follows that
Z
1
1
1
sup r̃t − AM(r̃t ) ≥
r̃t ωr̃nt ) ≥
(sup r̃t −
sup r̃t − C,
n+1 X
Vol(X) X
n+1 X
X
and this establishes the first estimate. The second estimate follows from (iv) and the
simple fact that supX r̃t ≥ AM(r̃t ).
Finally, we phrase some of the above estimates for AM–normalized Kähler–Ricci
trajectories:
Proposition 2.7. Suppose t → rt is a Kähler–Ricci trajectory that is AM–normalized,
i.e. AM(rt ) = 0. Let t → r̃t be the corresponding Kähler–Ricci trajectory normalized
according to (13) that corresponds to t → rt , i.e. rt = r̃t − AM(r̃t ). For t ≥ 0 the
following hold:
(i) − inf X rt ≤ C supX rt + C, for some C > 1.
(ii) supX r̃t ≤ C supX rt + C ≤ D supX r̃t + E, for some C, D, E > 1.
(iii) For any p ≥ 1 we have supX rt /C −C ≤ dp (r0 , rt ) ≤ C supX rt +C for some C > 1.
R
(iv) If α > n/(n+1) and p ≥ 1 then − log X e−α(rt −supX (rt −r0 ))+fω ω n ≤ −εdp (r0 , rt )+
C for some C > 1 and ε > 0.
Proof. The estimate in (i) follows from part (vi) of the previous proposition. This
last estimate also gives the first estimate of (ii). Estimate (ii) in the previous result
immediately gives the second part of (ii).
The first estimate of (iii) is just [Da4, Corollary 4]. By (12) we have that dp (r0 , rt ) ≤
oscX (r0 − rt ). Part (i) now implies the second estimate of (iii).
Notice that α > n/(n + 1) is equivalent with (1 − α)n − α < 0. The estimate of (iv)
now follows after we put together parts (v) of the previous proposition with what we
proved so far in this proposition.
3
Proof of the Main Results
First we give a proof for Theorem 1. As it turns out, the argument is about putting
together the pieces developed in the preceding sections.
9
Theorem 3.1. Suppose (X, J, ω) is a Fano manifold and p ≥ 1. There exists a Kähler–
Einstein metric in H if and only if every Kähler–Ricci trajectory [0, ∞) ∋ t → rt ∈ HAM
is dp –bounded.
Proof. If there exists a Kähler–Einstein metric in the cohomology class of ω then by
[Da4, Theorem 6] we have that any Kähler–Ricci trajectory dp –converges to one such
metric, hence stays dp –bounded.
For the other direction, suppose dp (0, rt ) is bounded. By Proposition 2.7(ii)(iii),
dp (0, rt ) controls supX r̃t , which in turn controls kr̃t kL∞ by Proposition 2.6(iv). The regularity theory for the Kähler–Ricci flow implies now that t → r̃t converges exponentially
fast in any C k norm to a Kähler–Einstein metric.
Theorem 3.2. Suppose (X, J, ω) is a Fano manifold without a Kähler–Einstein metric
in H and [0, ∞) ∋ t → rt ∈ HAM is a Kähler-Ricci trajectory. Then there exists a curve
[0, ∞) ∋ t → ut ∈ H0,AM which is a dp –geodesic ray weakly asymptotic to t → rt for all
p ≥ 1. In addition to this, t → ut satisfies the following:
(i) t → F (ut ) is decreasing,
(ii) the ”sup-normalized” potentials
unt − supX (ut − u0 ) ∈ H0 decrease pointwise to
R
u∞ ∈ PSH(X, ω) for which X e− n+1 u∞ ω n = ∞.
If additionally (X, J) does not admit non–trivial holomorphic vector fields then t → F (ut )
is strictly decreasing.
Proof. The idea of the proof is to construct a d2 –geodesic ray that satisfies all the
necessary properties. At the end we will conclude that this curve is also a dp –geodesic
ray for any p ≥ 1.
We can assume without loss of generality that r0 = 0. As a Kähler-Einstein metric
does not exist, by the previous theorem there exists tl → ∞ such that fl = d2 (0, rtl ) →
∞. Let [0, fl ] ∋ t → ult ∈ H∆ be the rescaled weak geodesic curve of (3), joining r0 = 0
with rtl . By our choice of normalization it follows that
AM(ult ) = 0 and d2 (0, ult ) = t, t ∈ [0, fl ].
(14)
Using Proposition 2.7(i) and (iii) there exists C, D > 1 such that
−Cd2 (0, rtl ) − C ≤ −D sup rtl − D ≤ inf rtl ≤ sup rtl ≤ Cd2 (0, rtl ) + C.
X
X
X
Rewriting this, as supX rtl → ∞, for l big enough we obtain:
−C ≤
−D ′ supX ulfl
inf X ulfl
supX ulfl
≤
≤
≤ C,
fl
fl
fl
(15)
for all fl ≥ 1. As ul0 = 0 for all l, using (15) and Theorem 2.5 we can conclude that
−C ≤
inf X ult
supX ult
−D ′ supX ult
≤
≤
≤ C, t ∈ [0, fl ].
t
t
t
(16)
By the results of [Brn1] and [BrmBrn] it also follows that the maps t → F (ult), M(ult )
are convex and non-positive. In particular, for t ≥ 0 we have:
Hω (ωult ) − L(ult )
t
=
M(ult) − M(u0 )
M(rtl ) − M(r0 )
≤ 0.
≤
t
fl
10
Proposition 2.4 now implies that there exists C > 1 such that
0 ≤ Hω (ωult ) ≤ L(ult ) ≤ Cd2 (0, ult) = Ct.
(17)
Fix now s ≥ 0. From (16) and (17) it follows using Theorem 2.3 that there exists lk′ → ∞
l′
and us ∈ H0 such that d2 (usk , us ) → 0. As AM is continuous with respect to d2 , by (14)
we also have AM(us ) = 0 and d2 (0, us) = s.
Building on this last observation, using a Cantor type diagonal argument, we can
find sequence lk → ∞ such that for each h ∈ Q+ there exists uh ∈ H0 satisfying
dp (ulhk , uh ) → 0, AM(uh ) = 0 and d2 (0, uh ) = h.
As t → ult are unit speed d2 –geodesic segments, it follows that for any a, b, c ∈ Q+
satisfying a < b < c we will also have
d2 (ua , ub ) + d2 (ub , uc ) = c − a = d2 (ua , uc ).
Hence, by density we can extend h → uh to a unit speed d2 –geodesic [0, ∞) ∋ t → ut ∈
H0,AM weakly asymptotic to t → rt . This dp –geodesic is non-trivial, i.e. not of the form
ut = u0 + ct for some c ∈ R. Indeed, this would contradict the fact AM(ut ) = 0 and
t → ut is unit speed with respect to d2 .
To show t → F (ut ) is decreasing, we claim first that for any t > 0, F (u0) ≥ F (ut).
First we note that F is continuous with respect to d2 . For each l, the map t → F (ult) is
convex and satisfies F (u0) ≥ F (ulfl ) hence for any t ∈ [0, fl ] we have F (u0) ≥ F (ult ). By
passing to the limit, the claim is proved. As t → F (ut) is convex and F (u0 ) ≥ F (ut ) for
any t ∈ (0, ∞), F has to be decreasing.
If additionally (X, J) does not admit non–trivial holomorphic vector fields then t →
F (ut ) is strictly decreasing. Indeed, if this were not the case, then there would exist
t0 ≥ 0 such that
∂
F (ut ) = 0, t ≥ t0 .
∂t
By Berndtsson’s convexity theorem [Brn1], this implies that (X, J) admits a non–trivial
holomorphic vector field, which is a contradiction.
We turn to part (ii). For n/(n + 1) < α < 1 each curve t → αult is a subgeodesic,
hence it follows from [Brn1] that each map
Z
l
t → − log
e−αut +fω ω n
X
is convex. As ul0 ≡ 0, by Theorem 2.5 the function
Z
Z
l
l
−α(ult −supX ult )+fω n
ω = − log
e−αut +fω ω n − α sup ult
t → Gα (ut ) = − log
e
X
X
X
is also convex. By theorem 2.7(iv) this implies that Gα (ult ) ≤ −εd2 (0, ult) + C = −εt + C.
Similarly to F (·), the functional Gα (·) is also continuous with respect to d2 , hence by
taking the limit lk → ∞ in this last estimate we obtain:
Gα (ut ) ≤ −εt + C.
(18)
As discussed after Theorem 2.5, the decreasing limit u∞ = limt→∞ (ut − supX ut ) is a
well defined and not identically equal to −∞. Letting t → ∞ in (18) we obtain that
11
e−αu∞ ω n = ∞. As n/(n+1) < α < 1, the recent resolution of the openness conjecture
(see [Brn2, GZh]) implies part (ii).
Finally, as t → ut is a bounded d2 –geodesic ray it follows by the discussion after (8)
that t → ut is a dp geodesic ray as well.
R
X
We believe t → F (ut ) should be strictly decreasing even if X has holomorphic vector
fields. We can show this when the Futaki invariant is nonzero as we elaborate below.
Note that along the Kähler-Ricci trajectory t → rt the F –functional is strictly decreasing
unless the initial metric is Kähler–Einstein. Using the identity
e−rt +fω
ω n = efωrt ωrnt
−rt +fω ω n
e
X
R
we can write
Z
∂F (rt )
=−
fωrt (efωrt − 1)ωrnt .
∂t
X
It is natural to introduce the following quanitity:
Z
ǫ(ω) = inf
fωu (efωu − 1)ωun ≥ 0.
u∈H
X
This quanitity is clearly an invariant of (X, J, [ω]). If ǫ(ω) > 0, then there
R existsn no
Kähler-Einstein metric in H. By Jensen’s inequality, for any u ∈ H we have M fωu ωu ≤
0, hence we can write
Z
Z
fω u
n
fω efωu ωun .
fωu (e − 1)ωu ≥
M
M
By [H2], the right hand side above (defined as the H-functional) is nonnegative and is
uniformly bounded away from zero if the Futaki invariant is nonzero, implying in this
last case the bound ǫ(ω) > 0. Finally, we note the following result:
Proposition 3.3. Suppose t → rt and t → ut are as in the previous theorem. If ǫ(ω) > 0,
then the map t → F (ut) is strictly decreasing. More precisely, there exists C > 0 such
that F (ut ) ≤ F (u0) − Ct, t ≥ 0.
Proof. By the discussion above, we have the estimate F (rtl ) − F (r0) ≤ −ǫ(ω)tl . Using
the notation of the previous theorem’s proof, by the estimates of Section 2.2, there exists
C, C ′ > 0 such that for l big enough:
fl = d2 (0, rtl ) ≤ C ′ sup rtl ≤ Ctl .
X
From our observations it follows that
F (ulfl ) − F (u0)
F (rtl ) − F (r0 )
ǫ(ω)
=
≤−
.
fl
fl
C
By the convexity of F we can conclude that
ǫ(ω)
F (ult ) − F (u0 )
≤−
, t ∈ (0, fl ].
t
C
Letting l → ∞ we obtain
F (ut ) − F (u0 )
ǫ(ω)
≤−
, t ∈ (0, ∞).
t
C
12
One would like to implement the ideas of this paper to construct geodesic rays asymptotic to other types of (geometric) flow trajectories. From the proof of Theorem 3.2 one
can extract the following general result:
Theorem 3.4. Suppose [0, ∞) ∋ t → ct ∈ HAM is a curve for which there exists tj → ∞
satisfying the following properties:
(i) There exists C > 1 such that − inf X ctj ≤ C supX ctj + C, t ≥ 0.
(ii) We have limj→∞ supX ctj = +∞ and
lim sup
j→∞
M(ctj ) − M(c0 )
< +∞.
supX ctj
Then there exists a curve [0, ∞) ∋ t → ut ∈ H0,AM which is a non–trivial dp –geodesic
ray weakly asymptotic to t → ct for all p ≥ 1.
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University of Maryland
[email protected]
University of Oregon
[email protected]
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